1. Statement of the Technical Field
The invention concerns methods and systems for demodulating Gaussian Filtered Minimum Shift Keyed (GMSK) signals, and more particularly concerns methods for demodulating GMSK signals having a low Bit period of Transmission (BT) value in receiver systems incorporating turbo equalization.
2. Description of the Related Art
The Radio Frequency (RF) bandwidth of a GMSK signal is advantageously controlled by using a Gaussian filter to shape the digital data stream prior to the digital data stream being applied to a phase modulator. GMSK is a type of phase modulation where the transmitted signal is given by the following mathematic equation (1).s(t,α)=exp(iψ(t,α))  (1)where t is the time. α is a vector of values defined by the following mathematical equation (2).αn=dndn−1  (2)where dn is the nth data symbol with a value of positive one (+1) or negative one (−1). ψ(t, α) is defined by the following mathematical equation (3).
                              ψ          ⁡                      (                          t              ,              α                        )                          =                              π            2                    ⁢                                    ∑              n                        ⁢                                          α                n                            ⁢                              q                ⁡                                  (                                      t                    -                    nT                                    )                                                                                        (        3        )            where q(t) is defined by the following mathematical equation (4).q(t)=∫−∞tg(τ)dτ  (4)where g(τ) is define by the following mathematical equation (5).
                              g          ⁡                      (            t            )                          =                              h            ⁡                          (              t              )                                *                      rect            ⁡                          (                              t                T                            )                                                          (        5        )            where * represents a convolution operation and h(t) is defined by the following mathematical equation (6).
                              h          ⁡                      (            t            )                          =                              exp            ⁡                          (                                                -                                      t                    2                                                                    2                  ⁢                                                                          ⁢                  δ                  ⁢                                                                          ⁢                                      T                    2                                                              )                                                                          2                ⁢                π                                      ⁢            δ            ⁢                                                  ⁢            T                                              (        6        )            where δ is defined by the following mathematical equation (7).
                    δ        =                                            ln              ⁡                              (                2                )                                                          2            ⁢            π            ⁢                                                  ⁢            BT                                              (        7        )            It is well known that sharper filters reduce unwanted sidelobes in the RF spectrum associated with GMSK signals, but also increases Inter-Symbol-Interference (ISI). Increased ISI causes symbol detection to be more difficult.
Error correcting codes do not function well where a high level of ISI is present in the communication channel. Accordingly, a receiver system can be designed to compensate for such channel effects prior to decoding. Such methods of compensation are typically referred to as channel equalization. In a system employing channel equalization, an equalizer is used to reduce the effects of ISI and other dispersive channel effects so as to increase the possibility of making correct decisions regarding which symbols have been received. Various methods for performing channel equalization are well known in the art, including the method known as turbo equalization.
It is known in the art that any constant amplitude binary phase modulation can be decomposed as a sum of a finite number of Pulse-Amplitude Modulated (PAM) signal components. See P. A. Laurent, “Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses”, IEEE Trans. Commun., vol. COMM-34 pp 150-160, February 1986. Such decomposition is sometimes referred to herein as a Laurent expansion. The terms of a Laurent expansion are signals in time, designated by k, that are given by the following mathematical equation (8).
                              ∑          n                ⁢                              b                          k              ,              n                                ⁢                                                    c                k                            ⁡                              (                                  t                  -                  nT                                )                                      .                                              (        8        )            where ck(t) represents pulses of the signals k.
An advantage of the decomposition or Laurent expansion is that, for some BT values, each signals k can be approximated using only one optimized amplitude modulated waveform (named a “main pulse”). For half-integer values of the modulation index, the main pulse is the first component of the decomposition or Laurent expansion, making computation of these functions particularly simple. These properties reduce significantly the inherent complexity of interpreting digital phase modulation, and therefore are especially useful for implementing demodulators.
High performance yet efficient trellis based demodulation, that can be represented using a single component of the above-referenced decomposition, have already been designed for GMSK applications where the value of BT is relatively high (BT≧0.25). For example, U.S. Pat. No. 6,993,070 to Berthet et al. discloses a turbo demodulation procedure for GMSK, but it is not applicable to GMSK with low BT values. Comparable algorithms for GMSK with significantly lower BT values that need to be represented by multiple terms in the Laurent expansion have not yet been developed. Current state of the art algorithms require the manipulation and book keeping of complex data values, and require separate data paths for odd and even samples. Furthermore, the trellis states are defined in terms of differentially encoded transmitted bits rather than actual data bits, making implementation of turbo equalization difficult. For example, See D. D. Abbey, R. R. Rhodes, “A simple coherent receiver for frequency-hopped pulse-driven GMSK”, IEEE 0-7803-5538-5/99, pp. 286-290, 1999.
GMSK is the modulation of choice for communications in space and for other applications. However, due to implementation concerns, GMSK has generally been considered only for use with BT values greater than one quarter (BT>0.25). Increases in data rates with reductions in transmission power are possible if suitable methods for implementation of lower BT GMSK can be developed. Such implementations are potentially very useful for systems requiring constant modulus transmission to reduce cost, that also need to minimize transmission power, for either cost or capacity concerns, but that can afford some degree of complexity at the receiver.